Jamming

1. Jamming Quantum Jamming in the ħ→ 0 limit Peter Olsson, Umeå University Stephen Teitel, University of Rochester Supported by: US Department of Energy Swedish High Performance Computing Center North

2. what is jamming? transition from flowing to rigid in condensed matter systems

3. cool Tm cool Tg the structural glass transition long range correlations short range correlations solid liquid shear stress solid: shear modulus liquid: shear viscosity ?????? correlations glass

4. the structural glass transition shear modulus shear viscosity shear modulus shear viscosity glass transition viscosity diverges glass: liquid: equilibrium transition? (diverging length scale) dynamic transition? (diverging time scale) no transition? (glass is just slow liquid) one of the greatest unresolved problems of condensed matter physics transition from flowing to rigid but disordered structure thermally driven

5. sheared foams polydisperse densely packed gas bubbles thermal fluctuations negligible critical yield stress foam has shear flow like a liquid foam ceases to flow and behaves like an elastic solid transition from flowing to rigid but disordered structure shear driven

6. critical volume density granular materials large weakly interacting grains thermal fluctuations negligible grains flow like a liquid grains jam, a finite shear modulus develops the jamming transition transition from flowing to rigid but disordered structure volume density driven

7. This false color image is taken from Dan Howell's experiments. This is a 2D experiment in which a collection of disks undergoes steady shearing. The red regions mean large local force, and the blue regions mean weak local force. The stress chains show in red. The key point is that on at least the scale of this experiment, forces in granular systems are inhomogeneous and itermittent if the system is deformed. We detect the forces by means of photoelasticity: when the grains deform, they rotate the polarization of light passing through them. Howell, Behringer, Veje, PRL 1999 and Veje, Howell, Behringer, PRE 1999

8. number of contacts: isostatic limit in d dimensions number of force balance equations: Nd (for repulsive frictionless particles) isostatic stability when these are equal Zis average contacts per particle seems well obeyed at jamming c

9. flowing ➝ rigid but disordered T surface below which states are jammed glass  yield stress J  conjecture by Liu and Nagel (Nature 1998) jamming, foams, glass, all different aspects of a unified phase diagram with three axes:  temperature   volume density   applied shear stress (nonequilibrium axis) “point J” is a critical point “the epitome of disorder” jamming transition “point J” critical scaling at point J influences behavior at finiteT and finite. understanding = 0 jamming at “point J” may have implications for understanding the glass transition at finite  here we consider the  plane at T = 0in 2D

10. ⇒ shear flow in fluid state shear stress  velocity gradient shear viscosity below jamming above jamming shear viscosity of a flowing granular material if jamming is like a critical point we expect

11. non-overlapping ⇒ non-interacting r (Durian, PRL 1995 (foams); O’Hern, Silbert, Liu, Nagel, PRE 2003) model granular material bidisperse mixture of soft disks in two dimensionsat T = 0 equal numbers of disks with diameters d1 = 1, d2 = 1.4 forN disks in area LxLy the volume density is interaction V(r)(frictionless) overlapping ⇒ harmonic repulsion overdamped dynamics

12. simulation parameters Lx = Ly N = 1024for < 0.844 N = 2048for ≥ 0.844 t ~ 1/N, integrate with Heun’s method total shear displacement ~ 10, ranging from 1 to 200 depending on N and finite size effects negligible (can’t get too close to c) animation at: = 0.830  0.838 < c≃ 0.8415 = 10-5

13. results for small = 10-5(represents → 0 limit, “point J”) as N increases,  vanishes continuously at c≃ 0.8415 smaller systems jam below c

14. c c results for finite shear stress 

15. critical “point J” control parameters  J ≡c ,   ,   c we thus get the scaling law scaling about “point J” for finite shear stress  scaling hypothesis(2nd order phase transitions): at a 2nd order critical point, a diverging correlation length determines all critical behavior quantities that vanish at the critical point all scale as some power of  rescaling the correlation length,  → b, corresponds to rescaling  ~ b1/ ,  ~ b ,  ~ b

16. crossover scaling function crossover scaling variable crossover scaling exponent  scaling law choose length rescaling factor b

17. scaling collapse of viscosity  point J is a true 2nd order critical point

18. transverse velocity correlation function (average shear flow along x)   distance to minimum gives correlation length  correlation length regions separated by  are anti-correlated motion is by rotation of regions of size 

19. scaling collapse of correlation length  diverges at point J

20. ' flowing shear stress  ' jammed  cz0 c 0 c volume density  “point J” phase diagram in  plane

21. conclusions • point J is a true 2nd order critical point • critical scaling extends to non-equilibrium driven steady states at finite shear stress   in agreement with proposal by Liu and Nagel • correlation length diverges at point J • diverging correlation length is more readily observed in driven non-equilibrium steady state than in equilibrium state • finite temperature?